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A sorting algorithm which uses the heap data structure

In computer science, heapsort is a comparison based sorting algorithm which can be thought of as "an implementation of selection sort using the right data structure." Like selection sort, heapsort divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element from it and inserting it into the sorted region. Unlike selection sort, heapsort does not waste time with a linear time scan of the unsorted region; rather, heap sort maintains the unsorted region in a heap data structure to efficiently find the largest element in each step. Although somewhat slower in practice on most machines than a well implemented quicksort, it has the advantages of very simple implementation and a more favorable worst case [[big O notation runtime. Most real world quicksort variants include an implementation of heapsort as a fallback should they detect that quicksort is becoming degenerate. Heapsort is an in place algorithm, but it is not a stable sort. Heapsort was invented by J. W. J. Williams in 1964. The paper also introduced the binary heap as a useful data structure in its own right. In the same year, Robert W. Floyd published an improved version that could sort an array in place, continuing his earlier research into the treesort algorithm.

Ternary heapsort uses a ternary heap instead of a binary heap; that is, each element in the heap has three children. It is more complicated to program, but does a constant number of times fewer swap and comparison operations. This is because each sift down step in a ternary heap requires three comparisons and one swap, whereas in a binary heap two comparisons and one swap are required. Two levels in a ternary heap cover 32 = 9 elements, doing more work with the same number of comparisons as three levels in the binary heap, which only cover 23 = 8. This is primarily of academic interest, or as a student exercise, as the additional complexity is not worth the minor savings, and bottom up heapsort beats both. Memory optimized heapsort improves heapsort's locality of reference by increasing the number of children even more. This increases the number of comparisons, but because all children are stored consecutively in memory, reduces the number of cache lines accessed during heap traversal, a net performance improvement. The standard implementation of Floyd's heap construction algorithm causes a large number of cache misses once the size of the data exceeds that of the CPU cache. Better performance on large data sets can be obtained by merging in depth first order, combining subheaps as soon as possible, rather than combining all subheaps on one level before proceeding to the one above. Out of place heapsort improves on bottom up heapsort by eliminating the worst case, guaranteeing n log2n + O(n) comparisons. When the maximum is taken, rather than fill the vacated space with an unsorted data value, fill it with a −∞ sentinel value, which never "bounces" back up. It turns out that this can be used as a primitive in an in place (and non recursive) "QuickHeapsort" algorithm. First, you perform a quicksort like partitioning pass, but reversing the order of the partitioned data in the array. Suppose (without loss of generality) that the smaller partition is the one greater than the pivot, which should go at the end of the array, but our reversed partitioning step places it at the beginning. Form a heap out of the smaller partition and do out of place heapsort on it, exchanging the extracted maxima with values from the end of the array. These are less than the pivot, meaning less than any value in the heap, so serve as −∞ sentinel values. Once the heapsort is complete (and the pivot moved to just before the now sorted end of the array), the order of the partitions has been reversed, and the larger partition at the beginning of the array may be sorted in the same way. (Because there is no non tail recursion, this also eliminates quicksort's O(log n) stack usage.) The smoothsort algorithm is a variation of heapsort developed by Edsger W. Dijkstra in 1981. Like heapsort, smoothsort's upper bound is O(n log n). The advantage of smoothsort is that it comes closer to O(n) time if the input is already sorted to some degree, whereas heapsort averages O(n log n) regardless of the initial sorted state. Due to its complexity, smoothsort is rarely used. Levcopoulos and Petersson describe a variation of heapsort based on a heap of Cartesian trees. First, a Cartesian tree is built from the input in O(n) time, and its root is placed in a 1 element binary heap. Then we repeatedly extract the minimum from the binary heap, output the tree's root element, and add its left and right children (if any) which are themselves Cartesian trees, to the binary heap. As they show, if the input is already nearly sorted, the Cartesian trees will be very unbalanced, with few nodes having left and right children, resulting in the binary heap remaining small, and allowing the algorithm to sort more quickly than O(n log n) for inputs that are already nearly sorted. Several variants such as weak heapsort require n log2 n+O(1) comparisons in the worst case, close to the theoretical minimum, using one extra bit of state per node. While this extra bit makes the algorithms not truly in place, if space for it can be found inside the element, these algorithms are simple and efficient, but still slower than binary heaps if key comparisons are cheap enough (e. g. integer keys) that a constant factor does not matter. Katajainen's "ultimate heapsort" requires no extra storage, performs n log2 n+O(1) comparisons, and a similar number of element moves. It is, however, even more complex and not justified unless comparisons are very expensive.

The examples sort the values { 6, 5, 3, 1, 8, 7, 2, 4 } in increasing order using both heap construction algorithms. The elements being compared are shown in a bold font. There are typically two when sifting up, and three when sifting down, although there may be fewer when the top or bottom of the tree is reached.